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Quenched invariance principle for simple random walk on clusters in correlated percolation models
Probability · Submitted 17 Oct 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z^d, d>=2, with long-range correlations introduced in arXiv:1212.2885, solving one of the open problems from there.
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The abstract, sentence by sentence
We prove quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z^d, d>=2, with long-range correlations introduced in arXiv:1212.2885, solving one of the open problems from there. This gives new results for random interlacements in dimension d>=3 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z^d, d>=2, with long-range correlations introduced in arXiv:1212.2885, solving one of the open problems from there. | contribution 0.58 | 0.95 | 0.95 | asserted 1.00 |
| s1 | This gives new results for random interlacements in dimension d>=3 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). | result 0.54 | 0.84 | 0.12 | asserted 0.97 |
| s2 | An essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models. | method 0.73 | 0.93 | 0.04 | asserted 1.00 |
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The paper
Original abstract
We prove quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z^d, d>=2, with long-range correlations introduced in arXiv:1212.2885, solving one of the open problems from there. This gives new results for random interlacements in dimension d>=3 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models.
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How the literature uses it
7 arXiv papers cite this in 24 sentences
Papers citing it · 18 sentences without a confident label
Uses its method or tool 1 paper
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Similar renormalization has been used in [ref] .
Random walks on infinite percolation clusters in models with long-range correlations · Some words about the proof of Theorem [ref] · 0.97
Cites it as background 4 papers
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Below we recall some results from [ref] about the large scale behavior of graph distances in $_∞$ and the quenched invariance principle for the simple random walk on $_∞$.
Random walks on infinite percolation clusters in models with long-range correlations · Known results · 0.96
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While for random interlacements itself many results have been shown to be valid for any $u > 0,$ the situation is more complicated when investigating the vacant set $𝒱^u.$ In fact, while there are few results concerning the vacant set in the first place so far, the ones which describe geometric properties such as Teixeira [ref] , Drewitz, Ráth and Sapozhnikov [ref] , Popov and Teixeira [ref] (dealing with the size distribution of finite clusters of the vacant set and local uniqueness properties of $𝒱_∞^u$) and Drewitz, Ráth and Sapozhnikov [ref] as well as Procaccia, Rosenthal and Sapozhnikov [ref] (providing chemical distance results as well as heat kernel estimates in a more general context) are valid for some non-degenerate fraction of the supercritical phase only.
Transience of the vacant set for near-critical random interlacements in high dimensions · Introduction · 0.84
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In a different direction, diffusions on clusters of correlated percolation models such as random interlacements are studied in [ref] via isoperimetric inequalities.
Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments · Introduction · 0.95
What it cites
Cites as background 1 paper
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In recent years interest arose in understanding such systems, both in specific models such as random interlacements, vacant set of random interlacements and the Gaussian free field, as well as in general systems, see for example [ref] .
Soft local times and decoupling of random interlacements · Introduction and results · 0.98
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